Bring a large pot of salted water to a boil and cook the lasagna noodles according to package directions; drain and set aside.
Preheat the oven to 350°F (175°C).
In a large skillet over medium heat, warm the vegetable oil. Add the chopped onion and garlic and cook until softened, about 3–5 minutes.
Add the ground beef and brown, breaking it up with a spoon, until no longer pink.
Stir in the canned tomatoes, tomato sauce, salt, pepper, basil, oregano, and sugar. Bring to a simmer, then cook uncovered for about 30 minutes to develop flavor.
Meanwhile, in a mixing bowl, beat the egg and stir in the cottage cheese until combined.
Grease a 2-quart casserole dish. Spread half of the meat sauce in the bottom, arrange 3 cooked lasagna noodles over the sauce, then spread half of the cottage cheese mixture and top with half of the sliced mozzarella.
Repeat the layers with the remaining sauce, 3 noodles, remaining cottage cheese mixture, and remaining mozzarella. Sprinkle the top with the Parmesan cheese.
Bake the assembled lasagna in the preheated oven for 30 to 40 minutes, until bubbly and heated through. Let rest a few minutes before serving.